A week and 3 major European capitals for February is a treat for the soul. This is only about transportation, not accommodation that you can easily book on yourself on http://www.booking.com or http://www.airbnb.com, particularly for Vienna as it is more expensive there. It is in fact a mixture between flights from and to Bucharest with Wizzair as they have a 20% discount promotion till 7th of January for the holders of card membership , and then buses with Flixbus that I check on the app GoEuro.
Why I like buses, and I used many time Flixbus, is that I avoid transfer times, expenses and hassle related to airports, and time of travel is under 4 hours usually between the major cities of Europe. The bus stops are in fact in bus or near trains stations in the city. Also prices are low, much more predictable than flights, and the times of departure and arrivals are more flexible.
Bucharest to Budapest on the 14th or the 12th of February at 8.2 euros.
Budapest to Vienna with Flixbus on the 16th of February, at 14 euros.
Vienna to Prague with RegioJet bus on the 18th of February, at 11.90 euros.
Prague to Nuremberg with Flixbus on the 22nd of February, at 14.90 euros
Nuremberg to Bucharest on the 24th or earlier on the 20th of February at 15.59 euros.
Now it is up to you to make your choice of dates, cause more available in more combinations, and enjoy.